A003355 -id:A003355 - cp's OEIS frontend

41, 103, 227, 283, 587, 829, 953, 1009, 1033, 1399, 1493, 1523, 1579, 1759, 2063, 2087, 2243, 2273, 2633, 2789, 2969, 3079, 3203, 3359, 3407, 3413, 3469, 3539, 3593, 3929, 4133, 4157, 4219, 4289, 4523, 4679, 4703, 5101, 5273, 5851, 6203, 6389, 6421, 6569, 6991

Offset: 1

Jonathan Vos Post, Sep 24 2006

Primes in the sumset {A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584 + A000584}.

There must be an odd number of odd terms in the sum, either one even and nine odd (as with 41 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 and 283 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5), three even and seven odd (as with 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 and 587 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5), five even and 5 odd terms (as with 17939 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 3^5 + 3^5 + 7^5), seven even and 3 odd terms (as with 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5) or nine even terms and one odd term (as with 3413 = 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 5^5). The sum of two positive 5th powers (A003347), other than 2 = 1^5 + 1^5, cannot be prime.

			a(1) = 41 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5.
a(2) = 103 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5.
a(3) = 227 = 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5.
a(4) = 283 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5.
a(5) = 587 = 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A000584, A003336, A003347, A003349, A003350, A003351, A003352, A003353, A003354, A003355.

up = 10^4; q = Range[up^(1/5)]^5; a = {0}; Do[b = Select[ Union@ Flatten@ Table[e + a, {e, q}], # <= up &]; a = b, {k, 10}]; Select[a, PrimeQ] (* Giovanni Resta, Jun 13 2016 *)

A000040 INTERSECTION A003355.

More terms from Alois P. Heinz, Aug 12 2015

cp's OEIS Frontend

A123043 Prime sums of 10 positive 5th powers.

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